The proof of Theorem REF  relies on the abstract scheme developed by Sandier & Serfaty in [1]}.
This scheme essentially consists of three main points.
First of all, we need that the Ginzburg-Landau energy, upon rescaling, \(\Gamma \) -converges to the renormalized energy \(W\) . As anticipated, this first point is provided by the recent results of Ignat & Jerrard ([2]}).
Second of all, we need a lower bound relating the properly rescaled \(L^2\)  norm of \(\partial _t u_\varepsilon \)  with the norm of the limit velocity \(\mathbf {a}^{\prime }\) .
Finally, we need a further lower bound relating the properly rescaled \(L^2\)  norm of
the gradient of the Ginzburg-Landau energy with the norm of the gradient of the renormalized energy.
As in the Euclidean case, the proof of the first lower bound relies on the so-called product estimates (see Sandier & Serfaty in [3]}) which are extended to our Riemannian setting.
The proof of the second lower bound relies on the expression for the gradient of the renormalized energy in terms of the canonical harmonic vector field, which is obtained as the limit of \(u_\varepsilon (t)\)  along a suitable subsequence of \(\varepsilon \) . This approach is in the spirit of known results in [4]} and [5]}, [6]}.
This strategy requires a careful study of the differentiability properties of the renormalized energy \(W\)  and of the structure of its gradient with respect to the position of the vortices.
